Stability of Reapproximation Algorithms for the $$\beta $$-Metric Traveling Salesman (Path) Problem

Abstract

Inspired by the concept of stability of approximation, we consider the following (re)optimization problem: Given a minimum-cost Hamiltonian cycle of a complete non-negatively real weighted graph G = (V, E, c) obeying the strengthened triangle inequality (i.e., for some strength factor 1/2 ≤ β < 1, we have that ∀u, v, z ∈ V, c(u, z) ≤ β(c(u, v) + c(v, z))), and given a vertex v whose removal from G (resp., addition to G), along with all its incident edges, produces a new weighted graph still obeying the strengthened triangle inequality, find a minimumcost Hamiltonian cycle of the modified graph. This problem is known to be NP-hard, but we show that it admits a PTAS, which just consists of either returning the old optimal cycle (after having bypassed the removed node), or instead computing (for finitely many inputs) a new optimal solution from scratch − depending on the required accuracy in the approximation. Then, we turn our attention to the case in which a minimum-cost Hamiltonian path is given instead, and the underlying graph obeys the relaxed triangle inequality. Here, if one edge weight is increased, and βO, βM ≥ 1 denotes the relaxation factor of the original and the modified graph, respectively, then we show how to obtain an approximation of 1 + 2 min{βO, βM}, which improves over existing solutions as soon as min{βO, βM} ≥ 3+2 √ 6 5